Towards analog implementations of PWL two‐dimensional non‐linear functions

This paper deals with the circuit implementation of non‐linear algebraic bivariate functions. The synthesis procedure is based on a piecewise‐linear approximation technique and on a corresponding circuit architecture, whose basic element is a circuit block with the input/output function y(x) = max(0; x). Some known CMOS circuit structures that can be used to obtain such a block are considered, and their main advantages and drawbacks are pointed out. The static and dynamic features of both the single circuit block and the overall architecture for two‐dimensional PWL functions are illustrated by way of examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Hybrid adaptive control for non‐linear uncertain impulsive dynamical systems

A direct hybrid adaptive control framework for non‐linear uncertain hybrid dynamical systems is developed. The proposed hybrid adaptive control framework is Lyapunov‐based and guarantees partial asymptotic stability of the closed‐loop hybrid system; that is, asymptotic stability with respect to part of the closed‐loop system states associated with the hybrid plant states. Furthermore, hybrid adaptive controllers guaranteeing attraction of the closed‐loop system plant states are also developed. Finally, two numerical examples are provided to demonstrate the efficacy of the proposed hybrid adaptive stabilization approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Constructive approximation of non‐linear discrete‐time systems

It is known that large classes of approximately‐finite‐memory maps can be uniformly approximated arbitrarily well by the maps of certain non‐linear structures. As an application, it was proved that time‐delay networks can be used to uniformly approximate arbitrarily well the members of a large class of causal nonlinear dynamic discrete‐time input–output maps. However, the proof is non‐constructive and provides no information concerning the determination of a structure that corresponds to a prescribed bound on the approximation error. Here we give some general results concerning the problem of finding the structure. Our setting is as follows. There is a large family 𝒢 of causal time‐invariant approximately‐finite‐memory input‐output maps G from a set S of real d‐vector‐valued discrete‐time inputs (with d⩾1) to the set of ℝ‐valued discrete‐time outputs, with both the inputs and outputs defined on the non‐negative integers 𝒵₊. We show that for each ϵ>0, any Gϵ𝒢 can be uniformly approximated by a structure map H(G, ·) to within tolerance ϵ, and we give analytical results and an example to illustrate how such a H(G, ·) can be determined in principle. Copyright © 2000 John Wiley & Sons, Ltd.     

Uniform approximation of time‐invariant non‐linear systems

An Erratum has been published for this article in International Journal of Circuit Theory and Applications 2004; 32(6):633. It is shown that the elements of a large class of time‐invariant non‐linear input–output maps can be uniformly approximated arbitrarily well, over infinite time intervals, using a certain structure that can be implemented in many ways using, for example, radial basis functions, polynomial functions, piecewise linear functions, sigmoids, or combinations of these functions. For the special case in which these functions are taken to be certain polynomial functions, the input–output map of our structure is a generalized finite Volterra series. Results are given for the case in which inputs and outputs are defined on ?. The case in which inputs and outputs are defined on the half‐line ?₊ is also addressed, and in both cases inputs need not be functions that are continuous. Copyright © 2004 John Wiley & Sons, Ltd.     

On simplex‐based piecewise‐linear approximations of non‐linear mappings

Simplex‐based piecewise‐linear (PWL) approximations of non‐linear mappings are needed when the robust PWL analysis is used to directly solve non‐linear equations. This paper proposes a straightforward technique for transforming the well‐known approximations into another form. This new form is computationally more efficient, since it preserves the sparse structure of the original Jacobian matrix. Furthermore, this new form of PWL approximation explicitly relates the simplex‐based PWL analysis to the conventional formulation of the Katzenelson algorithm. The proposed transform technique is also extended to treat groupwise‐separable mappings and, finally, non‐separable but sparse mappings that arise in real‐life simulation of large electronic circuits. In this paper, all these (transformed) simplex‐based PWL approximations are compared in terms of their generality and efficiency. The computational efficiency of the PWL approximation that utilizes sparsity is validated with realistic simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

A unified set‐theoretic approach to the analysis of PWL resistive circuits and composite N‐ports

We consider the search for all DC solutions of resistive piecewise‐linear (PWL) circuits and the analysis of the characteristics of resistive PWL composite N‐ports. These problems are unified from a theoretical and operating point of view by introducing the so‐called N‐augmented circuit, obtained from the N‐port by closing its ports with N norators. Set‐theoretic approach is used to describe the properties of the N‐augmented circuits leading to the formulation of a general DC analysis algorithm, based on linear programming techniques. The examples at the end of the paper show some practical and efficient application of the general DC analysis algorithm. Copyright © 2010 John Wiley & Sons, Ltd.     

Adaptive output tracking for a class of non‐linear time‐delay systems

In this paper the output tracking control problem for a class of non‐linear time delay systems with some unknown constant parameters is addressed. Such a problem is solved in the case that the non‐linear time‐delay system has full delay relative degree and stable internal dynamics. It is supposed moreover that the output and its time derivatives until n?1, where n is the length of the state vector (euclidean part), do not depend explicitly on the unknown parameters. This work is the first step towards the application of the methodologies of adaptive control for non‐linear delayless systems, based on tools of differential geometry, to non‐linear time‐delay systems too. Copyright © 2004 John Wiley & Sons, Ltd.     

On the terminal region of model predictive control for non‐linear systems with input/state constraints

This paper addresses the terminal region of model‐based predictive control (MPC) for non‐linear systems with control input and state constraints. Based on a stability condition of non‐linear MPC, a method to determine the terminal weighting term in the performance index and the terminal stabilizing control law to enlarge the terminal region and thus the domain of attraction of the non‐linear MPC is proposed. An LMI based optimization approach is developed to choose the terminal weighting item and fictitious terminal stabilizing control law so as to enlarge the terminal region of the non‐linear MPC method. The proposed method is illustrated by a numerical example and compares favourably with existing results. Copyright © 2003 John Wiley & Sons, Ltd.     

Closed‐loop iterative learning control for non‐linear systems with initial shifts

This paper is concerned with the problem of the iterative learning control with current cycle feedback for a class of non‐linear systems with well‐defined relative degree. The tracking error caused by a non‐zero initial shift is detected as extended D‐type learning algorithm is applied. The defect is overcome by adding terms including the output error, its derivatives as well as integrals. Asymptotic tracking of the final output to the desired trajectory is guaranteed. As an alternative approach, an initial rectifying action is introduced in the extended D‐type learning algorithm and shown effective to achieve the desired trajectory jointed smoothly with a transitional trajectory from the starting position. Also these algorithms with adjustable tracking interval ensure better robustness performance in the presence of initial shifts. Numerical simulation is conducted to demonstrate the theoretical results. Copyright © 2002 John Wiley & Sons, Ltd.     

Adaptive control for non‐negative and compartmental dynamical systems with applications to general anesthesia

Non‐negative and compartmental dynamical system models are composed of homogeneous interconnected subsystems or compartments which exchange variable non‐negative quantities of material with conservation laws describing transfer, accumulation, and elimination between the compartments and the environment. These models are widespread in biological and physiological sciences and play a key role in understanding these processes. In this paper, we develop a direct adaptive control framework for linear uncertain non‐negative and compartmental systems. The proposed framework is Lyapunov‐based and guarantees partial asymptotic set‐point regulation; that is, asymptotic set‐point stability with respect to part of the closed‐loop system states associated with the plant. In addition, the adaptive controller guarantees that the physical system states remain in the non‐negative orthant of the state space. Finally, a numerical example involving the infusion of the anesthetic drug propofol for maintaining a desired constant level of depth of anesthesia for non‐cardiac surgery is provided to demonstrate the efficacy of the proposed approach. Copyright © 2003 John Wiley & Sons, Ltd.     

A combined approach for the identification of continuous non‐linear systems

The identification of a non‐linear continuous output‐only system from a time series is considered for the case that the functional form of the model is not known beforehand. To estimate both functions and parameters, a combination of non‐parametric modelling based on non‐linear regression and parametric modelling based on a multiple shooting algorithm is proposed. This strategy to determine non‐linear differential equations is exemplified on experimental data from a chaotic circuit where an accurate reconstruction of the observed attractor is obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

Modeling of non‐linear systems by FWNNs and their intelligent control

By utilizing some of the important properties of wavelets like denoising, compression, multiresolution along with the concepts of fuzzy logic and neural network, two fuzzy wavelet neural networks (FWNNs) are proposed for approximating any arbitrary non‐linear function, hence, identifying a non‐linear system. We have fuzzified the output of DWT block, which receives the given inputs, in the proposed two methods: one using compression property and other using multiresolution property. We present a new type of fuzzy neuron model, each non‐linear synapse of which is characterized by a set of fuzzy implication rules with singleton weights in their consequents. It is shown that noise and disturbance in the reference signal are reduced with wavelets and also the variation of somatic gain, the parameter that controls the slope of the activation function in the neural network, leads to more accurate output. Identification results are found to be accurate and speed of their convergence is fast. Next, we simulate a control system for keeping output at a desired level by using the identified models. Two self‐learning controllers are designed in this simulation. One is a self‐learning fuzzy PI controller and other is a NN controller. Simulation results show that the NN controller is more adaptive and fast. Copyright © 2005 John Wiley & Sons, Ltd.     

Threshold computation for fault detection in linear discrete‐time Markov jump systems

This paper proposes a threshold computation scheme for an observer‐based fault detection (FD) in linear discrete‐time Markovian jump systems. An observer‐based FD scheme typically consists of two stages known as residual generation and residual evaluation. Even information of faults is contained inside a residual signal, a decision of faults occurrence is consequently made by a residual evaluation stage, which consists of residual evaluation function and threshold setting. For this reason, a successful FD strongly depends on a threshold setting for a given residual evaluation function. In this paper, Kalman filter (KF) is used as a residual generator. Based on an accessibility of Markov chain to KF, two types of residual generations are considered, namely mode‐dependent and mode‐independent residual generation. After that threshold is computed in a residual evaluation stage such that a maximum fault detection rate is achieved, for a given false alarm rate. Without any knowledge of a probability density function of residual signal before and after fault occurrence, a threshold is computed by using an estimation of residual evaluation function variance in a fault‐free case. Finally, a detection performance is demonstrated by a numerical example. Copyright © 2013 John Wiley & Sons, Ltd.     

Direct adaptive control for non‐linear uncertain systems with exogenous disturbances

A direct adaptive non‐linear control framework for multivariable non‐linear uncertain systems with exogenous bounded disturbances is developed. The adaptive non‐linear controller addresses adaptive stabilization, disturbance rejection and adaptive tracking. The proposed framework is Lyapunov‐based and guarantees partial asymptotic stability of the closed‐loop system; that is, asymptotic stability with respect to part of the closed‐loop system states associated with the plant. In the case of bounded energy L₂ disturbances the proposed approach guarantees a non‐expansivity constraint on the closed‐loop input–output map. Finally, several illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach. Copyright © 2002 John Wiley & Sons, Ltd.     

Adaptive estimation and rejection of unknown sinusoidal disturbances through measurement feedback for a class of non‐minimum phase non‐linear MIMO systems

This paper develops an adaptive estimation method to estimate unknown disturbances in a class of non‐minimum phase non‐linear MIMO systems. The unknown disturbances are generated by an unknown linear exosystem. The frequencies, phases and amplitudes of the disturbances are unknown, the only available information of the disturbances is the number of distinctive frequencies. The system considered in this paper is a class of MIMO non‐linear systems in the output feedback form which can be non‐minimum phase. The proposed estimation algorithm provides exponentially convergent estimates of system states, unknown disturbances in the system and frequencies of the disturbances characterized by the eigenvalues of the exosystem. Moreover, based on the stabilization controller for the disturbance free system, the estimates of the disturbances are used to solve the disturbance rejection problem. The unknown disturbances are compensated completely with the stability of the whole closed‐loop system. Copyright © 2006 John Wiley & Sons, Ltd.     

Experimentally verified point‐to‐point iterative learning control for highly coupled systems

Iterative learning control (ILC) is a well‐established approach for precision tracking control of systems, which perform a repeated tracking task defined over a fixed time interval. Despite a rich theoretical framework accompanied by a wide array of application studies, comparatively little attention has been paid to the case of multiple input, multiple output (MIMO) systems. Here, the presence of interacting dynamics often correlates with reduced performance. This article focuses on a general class of linear ILC algorithms and establishes links between interaction dynamics and reduced robustness to modeling uncertainty, and slower convergence. It then shows how these and other limitations can be addressed by relaxing the tracking requirement to include only a subset of time points along the time duration. This is the first analysis to show how so‐called ‘point‐to‐point’ ILC can address performance limitations associated with highly coupled systems. Theoretical observations are tested using a novel MIMO experimental test facility, which permits both exogenous disturbance injection and a variable level of coupling between input and output pairs. Results compare experimental observations with theoretical predictions over a wide range of interaction levels and with varying levels of injected noise. Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive control of non‐linear systems with unknown virtual control coefficients

This paper deals with state feedback adaptive control of parametric‐strict‐feedback (triangular) non‐linear systems with unknown virtual control coefficients. A priori knowledge of the signs of the virtual coefficients is not required, and control signals and adaptive laws are smooth. Asymptotic tracking of smooth reference signals is achieved while all the variables remain bounded. The proposed algorithms make use of backstepping and tuning functions, and enlarge the class of non‐linear systems with unknown parameters for which asymptotic output tracking can be achieved. Copyright © 2000 John Wiley & Sons, Ltd.     

State controllability computation technique for linear time‐varying systems by using Taylor series approximation

This article presents a technique to determine the controllability Grammian matrix (CGM) for linear time‐varying systems by using truncated Taylor polynomial vector and the operational matrix of integration. An important property of this algorithm is that it starts by integrating the Lyapunov differential matrix equation in terms of the CGM. However, the algorithm does not use the mathematical integration processes actually, but uses the truncated Taylor polynomial vector and the operational matrix of integration. Thus, the problem is reduced to solving a linear set of algebraic equations with constant coefficients consisting of the Taylor polynomial constant coefficients of each of the CGM elements. Numerical results and error curves are given to illustrate the improvements achieved by the proposed algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

A method for the approximate synthesis of cellular non‐linear networks—Part 2: Circuit reduction

In this paper, we face the problem of model reduction in piecewise‐linear (PWL) approximations of non‐linear functions. The reduction procedure presented here is based on the PWL approximation method proposed in a companion paper and resorts to a strategy that exploits the orthonormality of basis functions in terms of a proper inner product. Such a procedure can be favourably applied to the synthesis of the resistive parts of cellular non‐linear networks (CNNs) to reduce the complexity of the resulting circuits. As an example, the method is applied to a case study concerning a CNN for image processing. Copyright © 2003 John Wiley & Sons, Ltd.